Homology equivalences inducing an epimorphism on the fundamental group and Quillen's plus construction

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing complex $D$. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated sc…

A discrete model of $S^1$-homotopy theory

We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between $\sP$ and the model category of $S^1$-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.

Topological model categories generated by finite complexes

Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -- describing ho…

非球面空间的序列拓扑复杂度与子群包含的截面范畴

Sequential topological complexity of aspherical spaces and sectional categories of subgroup inclusions

本文将非球面空间TC的拓扑机器人学结果推广至诱导子群包含的纤维化截面范畴，给出序列TC与满同态参数化TC的新下界，并推广Costa-Farber典范类以研究更一般空间。

Relative plus constructions

Let $h$ be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair $(X, H)$ consisting of a connected space $X$ and an $h$-perfect normal subgroup $H$ of the fundamental group $π_1(X)$ an $h$-acyclic map $X \rightarrow X^{+h}_H$ inducing the quotient by $H$ on the fundamental group. When $h$ is an ordinary homology theory with coefficients in a commutative ring with unit $R$, this provides a functorial and well-defined counterpart to a construction by cell attac…

三种弱等价概念下的链的单纯余代数

The simplicial coalgebra of chains under three different notions of weak equivalence

本文针对三种弱等价概念，在约化单纯集和连通单纯余交换余代数上构造模型结构，并证明链的单纯余代数在代数闭域上定义同伦完全忠实的左 Quillen 函子，在完美域上通过同伦不动点比较各模型结构。